Integrand size = 23, antiderivative size = 52 \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=-\frac {(a-b) \arctan \left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2} d}+\frac {\sinh (c+d x)}{b d} \] Output:
-(a-b)*arctan(b^(1/2)*sinh(d*x+c)/a^(1/2))/a^(1/2)/b^(3/2)/d+sinh(d*x+c)/b /d
Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.96 \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {-\frac {(a-b) \arctan \left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}+\frac {\sinh (c+d x)}{b}}{d} \] Input:
Integrate[Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]^2),x]
Output:
(-(((a - b)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(3/2))) + Sinh[c + d*x]/b)/d
Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 3669, 299, 218}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\cosh ^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (i c+i d x)^3}{a-b \sin (i c+i d x)^2}dx\) |
\(\Big \downarrow \) 3669 |
\(\displaystyle \frac {\int \frac {\sinh ^2(c+d x)+1}{b \sinh ^2(c+d x)+a}d\sinh (c+d x)}{d}\) |
\(\Big \downarrow \) 299 |
\(\displaystyle \frac {\frac {\sinh (c+d x)}{b}-\frac {(a-b) \int \frac {1}{b \sinh ^2(c+d x)+a}d\sinh (c+d x)}{b}}{d}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\frac {\sinh (c+d x)}{b}-\frac {(a-b) \arctan \left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}}{d}\) |
Input:
Int[Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]^2),x]
Output:
(-(((a - b)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(3/2))) + Sinh[c + d*x]/b)/d
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f S ubst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x] /ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Time = 10.22 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {\frac {\sinh \left (d x +c \right )}{b}+\frac {\left (-a +b \right ) \arctan \left (\frac {b \sinh \left (d x +c \right )}{\sqrt {a b}}\right )}{b \sqrt {a b}}}{d}\) | \(44\) |
default | \(\frac {\frac {\sinh \left (d x +c \right )}{b}+\frac {\left (-a +b \right ) \arctan \left (\frac {b \sinh \left (d x +c \right )}{\sqrt {a b}}\right )}{b \sqrt {a b}}}{d}\) | \(44\) |
risch | \(\frac {{\mathrm e}^{d x +c}}{2 b d}-\frac {{\mathrm e}^{-d x -c}}{2 b d}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right ) a}{2 \sqrt {-a b}\, d b}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{2 \sqrt {-a b}\, d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right ) a}{2 \sqrt {-a b}\, d b}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{2 \sqrt {-a b}\, d}\) | \(193\) |
Input:
int(cosh(d*x+c)^3/(a+b*sinh(d*x+c)^2),x,method=_RETURNVERBOSE)
Output:
1/d*(1/b*sinh(d*x+c)+(-a+b)/b/(a*b)^(1/2)*arctan(b*sinh(d*x+c)/(a*b)^(1/2) ))
Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (44) = 88\).
Time = 0.14 (sec) , antiderivative size = 662, normalized size of antiderivative = 12.73 \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx =\text {Too large to display} \] Input:
integrate(cosh(d*x+c)^3/(a+b*sinh(d*x+c)^2),x, algorithm="fricas")
Output:
[1/2*(a*b*cosh(d*x + c)^2 + 2*a*b*cosh(d*x + c)*sinh(d*x + c) + a*b*sinh(d *x + c)^2 + sqrt(-a*b)*((a - b)*cosh(d*x + c) + (a - b)*sinh(d*x + c))*log ((b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^ 4 - 2*(2*a + b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 - 2*a - b)*sinh(d *x + c)^2 + 4*(b*cosh(d*x + c)^3 - (2*a + b)*cosh(d*x + c))*sinh(d*x + c) - 4*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 - 1)*sinh(d*x + c) - cosh(d*x + c))*sqrt(-a*b) + b)/(b *cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 + 2*(2*a - b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 + 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + b )) - a*b)/(a*b^2*d*cosh(d*x + c) + a*b^2*d*sinh(d*x + c)), 1/2*(a*b*cosh(d *x + c)^2 + 2*a*b*cosh(d*x + c)*sinh(d*x + c) + a*b*sinh(d*x + c)^2 + 2*sq rt(a*b)*((a - b)*cosh(d*x + c) + (a - b)*sinh(d*x + c))*arctan(2*sqrt(a*b) /(b*cosh(d*x + c) + b*sinh(d*x + c))) - 2*sqrt(a*b)*((a - b)*cosh(d*x + c) + (a - b)*sinh(d*x + c))*arctan(1/2*(b*cosh(d*x + c)^3 + 3*b*cosh(d*x + c )*sinh(d*x + c)^2 + b*sinh(d*x + c)^3 + (4*a - b)*cosh(d*x + c) + (3*b*cos h(d*x + c)^2 + 4*a - b)*sinh(d*x + c))*sqrt(a*b)/(a*b)) - a*b)/(a*b^2*d*co sh(d*x + c) + a*b^2*d*sinh(d*x + c))]
Timed out. \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cosh(d*x+c)**3/(a+b*sinh(d*x+c)**2),x)
Output:
Timed out
\[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\int { \frac {\cosh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right )^{2} + a} \,d x } \] Input:
integrate(cosh(d*x+c)^3/(a+b*sinh(d*x+c)^2),x, algorithm="maxima")
Output:
1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)/(b*d) - 1/8*integrate(16*((a*e^(3*c ) - b*e^(3*c))*e^(3*d*x) + (a*e^c - b*e^c)*e^(d*x))/(b^2*e^(4*d*x + 4*c) + b^2 + 2*(2*a*b*e^(2*c) - b^2*e^(2*c))*e^(2*d*x)), x)
Exception generated. \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(cosh(d*x+c)^3/(a+b*sinh(d*x+c)^2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Time = 2.01 (sec) , antiderivative size = 426, normalized size of antiderivative = 8.19 \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {{\mathrm {e}}^{c+d\,x}}{2\,b\,d}-\frac {\left (2\,\mathrm {atan}\left (\frac {a\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {4\,\left (2\,a\,b^3\,d\,\sqrt {a^2-2\,a\,b+b^2}+2\,a^3\,b\,d\,\sqrt {a^2-2\,a\,b+b^2}-4\,a^2\,b^2\,d\,\sqrt {a^2-2\,a\,b+b^2}\right )}{a^2\,b^7\,d^2\,\left (a-b\right )}-\frac {2\,\left (a^3\,\sqrt {a\,b^3\,d^2}-b^3\,\sqrt {a\,b^3\,d^2}+3\,a\,b^2\,\sqrt {a\,b^3\,d^2}-3\,a^2\,b\,\sqrt {a\,b^3\,d^2}\right )}{a^2\,b^5\,d\,\sqrt {{\left (a-b\right )}^2}\,\sqrt {a\,b^3\,d^2}}\right )\,\sqrt {a\,b^3\,d^2}}{4\,a^2-8\,a\,b+4\,b^2}+\frac {2\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\left (a^3\,\sqrt {a\,b^3\,d^2}-b^3\,\sqrt {a\,b^3\,d^2}+3\,a\,b^2\,\sqrt {a\,b^3\,d^2}-3\,a^2\,b\,\sqrt {a\,b^3\,d^2}\right )}{a\,b\,d\,\sqrt {{\left (a-b\right )}^2}\,\left (4\,a^2-8\,a\,b+4\,b^2\right )}\right )+2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a-b\right )\,\sqrt {a\,b^3\,d^2}}{2\,a\,b\,d\,\sqrt {{\left (a-b\right )}^2}}\right )\right )\,\sqrt {a^2-2\,a\,b+b^2}}{2\,\sqrt {a\,b^3\,d^2}}-\frac {{\mathrm {e}}^{-c-d\,x}}{2\,b\,d} \] Input:
int(cosh(c + d*x)^3/(a + b*sinh(c + d*x)^2),x)
Output:
exp(c + d*x)/(2*b*d) - ((2*atan((a*b^4*exp(d*x)*exp(c)*((4*(2*a*b^3*d*(a^2 - 2*a*b + b^2)^(1/2) + 2*a^3*b*d*(a^2 - 2*a*b + b^2)^(1/2) - 4*a^2*b^2*d* (a^2 - 2*a*b + b^2)^(1/2)))/(a^2*b^7*d^2*(a - b)) - (2*(a^3*(a*b^3*d^2)^(1 /2) - b^3*(a*b^3*d^2)^(1/2) + 3*a*b^2*(a*b^3*d^2)^(1/2) - 3*a^2*b*(a*b^3*d ^2)^(1/2)))/(a^2*b^5*d*((a - b)^2)^(1/2)*(a*b^3*d^2)^(1/2)))*(a*b^3*d^2)^( 1/2))/(4*a^2 - 8*a*b + 4*b^2) + (2*exp(3*c)*exp(3*d*x)*(a^3*(a*b^3*d^2)^(1 /2) - b^3*(a*b^3*d^2)^(1/2) + 3*a*b^2*(a*b^3*d^2)^(1/2) - 3*a^2*b*(a*b^3*d ^2)^(1/2)))/(a*b*d*((a - b)^2)^(1/2)*(4*a^2 - 8*a*b + 4*b^2))) + 2*atan((e xp(d*x)*exp(c)*(a - b)*(a*b^3*d^2)^(1/2))/(2*a*b*d*((a - b)^2)^(1/2))))*(a ^2 - 2*a*b + b^2)^(1/2))/(2*(a*b^3*d^2)^(1/2)) - exp(- c - d*x)/(2*b*d)
Time = 0.18 (sec) , antiderivative size = 807, normalized size of antiderivative = 15.52 \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx =\text {Too large to display} \] Input:
int(cosh(d*x+c)^3/(a+b*sinh(d*x+c)^2),x)
Output:
(2*e**(c + d*x)*sqrt(b)*sqrt(a)*sqrt(a - b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2 *a - b)*atan((e**(c + d*x)*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)))*a - 2*e**(c + d*x)*sqrt(b)*sqrt(a)*sqrt(a - b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)*atan((e**(c + d*x)*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)))*b - 2*e**(c + d*x)*sqrt(b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)*atan((e**(c + d*x)*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b))) *a**2 + 2*e**(c + d*x)*sqrt(b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)*atan( (e**(c + d*x)*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)))*a*b + e* *(c + d*x)*sqrt(b)*sqrt(a)*sqrt(a - b)*sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b)*log( - sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b) + e**(c + d*x)*sqrt(b))*a - e**(c + d*x)*sqrt(b)*sqrt(a)*sqrt(a - b)*sqrt(2*sqrt(a)*sqrt(a - b) - 2* a + b)*log( - sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b) + e**(c + d*x)*sqrt(b) )*b - e**(c + d*x)*sqrt(b)*sqrt(a)*sqrt(a - b)*sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b)*log(sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b) + e**(c + d*x)*sqrt(b ))*a + e**(c + d*x)*sqrt(b)*sqrt(a)*sqrt(a - b)*sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b)*log(sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b) + e**(c + d*x)*sqrt( b))*b + e**(c + d*x)*sqrt(b)*sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b)*log( - sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b) + e**(c + d*x)*sqrt(b))*a**2 - e**(c + d*x)*sqrt(b)*sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b)*log( - sqrt(2*sqrt(a )*sqrt(a - b) - 2*a + b) + e**(c + d*x)*sqrt(b))*a*b - e**(c + d*x)*sqr...